发表机构
Wrocław University of Science and Technology（弗罗茨瓦夫理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对复值信号建模，提出圆柱流匹配（CyFM），通过解耦度量与精确最优传输耦合，显著降低少步生成误差，并揭示因子化耦合陷阱。
AI 中文摘要
复值信号，例如磁共振成像（MRI）和音频频谱图，几乎总是被建模为平坦的双通道欧几里得数据。对于非零值，幅度-相位图 $z \mapsto (|z|, z/|z|)$ 将信号域识别为圆柱体 $(0, \infty) \times S^1$，在该圆柱体上，我们有意将继承的度量 $dA^2 + A^2 d\theta^2$ 替换为解耦的乘积度量 $dA^2 + d\theta^2$。在这项实证研究中，我们衡量了这种替换的代价与收益。通过计算精确的解析桥，我们证明笛卡尔路径会引发角速度的重尾分布（幂律指数 $\approx 1.0$），在独立耦合下，近一半的概率路径的角速度超过 $\pi$，而圆柱路径永远不会超过该速率。为解决此问题，我们分析了圆柱流匹配（CyFM），它严格限制回归目标，并通过在圆柱度量下对整个场联合计算的精确小批量最优传输来耦合噪声和数据。尽管这种耦合的传输成本降低随场维度而崩溃（从标量对的86%降至$64\times64$场的3%），但其对少步生成的益处并未消失：它在每个评估分辨率下将圆柱模型的少步误差降低了3-60%。借助这种耦合，CyFM在每一步数（直至$k=8$）和每个评估分辨率下，其误差均低于最佳笛卡尔基线，且所有五个种子均分离，无需蒸馏；在收敛时，我们未检测到两种几何之间的显著差异。最后，我们揭示了“因子化耦合陷阱”，表明逐维度或逐块的传输因子化会悄然破坏数据的联合分布。所有实验均在合成复值场上进行。
英文摘要
Complex-valued signals like MRI and audio spectrograms are typically modelled as flat two-channel Euclidean data. The inherited Euclidean metric $dA^2 + A^2 dθ^2$ vanishes at the origin, leaving phase unpenalised exactly where the signal is weakest. We replace it with the decoupled product metric $dA^2 + dθ^2$ on the cylindrical closure $[0, \infty) \times S^1$, which stays non-degenerate at $A = 0$. We measure what this substitution costs and buys. Exact analytical bridges across synthetic fields, fastMRI knee data, and LibriSpeech spectrograms show Cartesian paths induce a heavy-tailed angular velocity distribution (Pareto index $\approx 1$). Under independent coupling, 43%-49% of signal energy falls on paths turning faster than $π$ rad per unit time. Cylindrical paths never reach this speed. We formulate Cylindrical Flow Matching (CyFM) to strictly bound the angular regression target, coupling noise and data via exact minibatch Optimal Transport jointly over whole fields. This coupling reduces few-step generation error by 3%-60%. CyFM achieves lower generative error than the best Cartesian baseline at every step up to $k = 8$ on synthetic fields and speech spectrograms, with all seeds separated. On knee MRI, the single-step advantage is 1.8x. At convergence ($k = 100$), the two geometries show no significant difference. Finally, a prior-only control exposes the cost of flat parametrisation: on synthetic fields, a single Cartesian Euler step performs worse than the unintegrated noise prior (0.376 vs. 0.150).
CommentsPreprint. 19 pages, 2 figures, 3 tables
